| Résume||The aim of the lecture is to show that Aut(Fₙ), the automorphism group of the free group has Kazdhan property (T). I will begin by studying the structure of Aut(Fₙ) and the action of its "Weyl group". Then I will show you how to square the group Laplacian in a form that is compatible with the action. This description can be then used to define operators Adjₙ and Opₙ and reformulate property (T) in terms of their positivity. Finally I will show to use the Weyl group to reduce property (T) for Aut(Fₙ) for any n ≥ 5 to a sum of squares problem in ℝ[Aut(F₅)] which can be effectively checked on a computer.
The talk will be based on 1812.03456 by D.Kielak, P.Nowak and myself.|